Complete group classification of a class of nonlinear wave equations

Abstract

Preliminary group classification became prominent as an approach to symmetry analysis of differential equations due to the paper by Valenti [J. Math. Phys. 32(11), 2988-2995] in which partial preliminary group classification of a class of nonlinear wave equations was carried out via the classification of one-dimensional Lie symmetry extensions related to a fixed finite-dimensional subalgebra of the infinite-dimensional equivalence algebra of the class under consideration. In the present paper we implement, up to both usual and general point equivalence, the complete group classification of the same class using the algebraic method of group classification. This includes the complete preliminary group classification of the class and finding Lie symmetry extensions which are not associated with subalgebras of the equivalence algebra. The complete preliminary group classification is based on listing all inequivalent subalgebras of the whole infinite-dimensional equivalence algebra whose projections are qualified as maximal extensions of the kernel algebra. The set of admissible point transformations of the class is exhaustively described in terms of the partition of the class into normalized subclasses. A version of the algebraic method for finding the complete equivalence groups of a general class of differential equations is proposed.

Complete point symmetry group of the barotropic vorticity equation on a rotating sphere

“…The same transformation also allows one to set Ω to zero in the vorticity equation (1). Note that the transformation (2) was originally derived in [14], where it was used to transform the vorticity equation into a reference frame with vanishing angular momentum.…”

“…We simplify the computation within the framework of the direct method by combining it with an advanced version of the algebraic approach originally proposed in [8,9], essentially modified in [4] and then developed in [1,3]. As a result, we prove that in fact the group G Ω is generated by Lie symmetry transformations of the sBVE and the previously mentioned two discrete transformations.…”

Group classification of linear Schrödinger equations by the algebraic method

“…In this section we look at equivalence transformations of classes of differential equations and some of their properties. More details can be found in [3,13,24].…”

“…It simplifies the process of classifying non trivial Lie symmetries within this class. More details on the use of the above approaches can be found in [3,10,8,7], where the problem of the group classification of nonlinear Schrödinger equations is considered.…”

“…From the early 1960's onwards Lie symmetry analysis of differential equations became an area of intensive research producing many new results (although quite a few of these had in fact been discovered by Lie). Now there are many textbooks and research papers on the subject (see for instance [3,21,4,9,20]) and the Lie approach has become quite a standard method.…”